Center of inertia and coordinate transformations in the post-Newtonian charged n-body problem in gravitation

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Abstract
We generalize the field theory propagator by finding a way to make it a function of some additional arbitrary parameters. Thus, it is now possible to obtain Lagrangians (which contain the propagator parameters) from field theory in a more general coordinate system than had previously been possible. We find the n‐body (classical) Bażański Lagrangian in this more general coordinate system and we give the relationship between the various coordinate systems by an n‐body coordinate transformation involving the propagator parameters. We find the center of inertia for the case of the n‐body Basżański Lagrangian in the general coordinate system and find that the potential energy terms −Gmimj/rij and eiej/rij do not in general split equally between particles i and j as they do in the case of Bażański coordinates. We also find the center of inertia for the case of the n‐body (unchanged) post‐Newtonian Lagrangian with parameterized post‐Newtonian (PPN) parameters γ and β in standard coordinates, and show that the potential energy terms do split equally between a pair of particles.